Future high-speed and large-capacity optical communication will advance toward coherent optical communication. Various modulation formats, for example, amplitude modulation, phase modulation, orthogonal amplitude modulation (QAM), etc., can be used in a coherent optical communication system. Phase shift keying (PSK) is a format of pure phase modulation in which information to be transported is represented with the phase of a symbol. PSK with a number m of possible symbols for transmission is referred to as mPSK (m is an integer larger than or equal to 2) in which a symbol S can be represented as S=exp(j·2kπ/m), where k=0, 1, . . . m−1, and the symbol S takes a value varying with k. General modulation formats of BPSK (binary phase shift keying), QPSK (orthogonal phase shift keying or quarter phase shift keying), etc., each are specific examples of mPSK with m taking values of 2 and 4 respectively.
Since inter-symbol interference (ISI) may be introduced into an optical channel, ISI would be removed in a coherent optical receiver typically using adaptive equalization technique based upon a finite impulse response (FIR) filter. Also blind equalization is desirably used to improve the utilization ratio of the spectrum and avoid an extra overhead due to training sequences. The constant modulus algorithm (CMA) is one of the most common adaptive blind equalization algorithms, e.g., Blind equalization using the constant modulus criterion: A Review by C, Richard Johnson, et al, in Proceedings of the IEEE, vol. 86, no. 10, October 1998, may be cited, the disclosure of which is incorporated here by reference in its entirety.
The CMA is applicable to the predominant modulation format QPSK adopted in the current coherent communication system. Unfortunately, one of perquisites for normal operation of the CMA is a circularly symmetric constellation diagram of a signal, i.e., E{S2}=0 (that is, the expectation of S2 is 0), where S denotes a symbol (for example, S=exp(jk2π/m) for an mPSK signal). See Adaptive Filter Theory, 3rd edition by Simon Haykin (Prentice Hall, 1998), for example, the disclosure of which is incorporated here by reference in its entirety. Except for a BPSK signal, all the mPSK signals with m>2 satisfy this condition. This means inapplicability of the CMA to the BPSK signal. See On the existence of undesirable global minimum of Godard equalizers by C. B. Papadias in Acoustics, Speech and Signal Processing, vol. 5, 1997, the disclosure of which is incorporated here by reference in its entirety.